Honors Math 3
Name:
Date:
Central Limit Theorem
Warm-up
Consider spinner with mean m and standard deviation s . If the spinner is spun n times
and the results added, how do the mean and standard deviation change?
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Exploration
1. A spinner has the number 1, 3, 5 and 7 on it. What is the mean and standard deviation
for one spin? Leave your standard deviation as a radical.
2. The spinner will be spun twice and the results are averaged. The following table gives
the sample space for the possible averages. What is the mean and standard deviation
for this sample space? Again, leave your standard deviation as a radical.
average
1
3
5
7
1
3
5
7
1
2
3
4
2
3
4
5
3
4
5
6
4
5
6
7
3. How did the mean and standard deviation change from one spin to averaging two
spins?
Generalize
Consider spinner with mean m and standard deviation s . If the spinner is spun n times
and the results averaged, how do the mean and standard deviation change?
Central Limit Theorem: Let X be a random variable with mean m and standard
deviation s . The distribution for the sum of the outputs of X over n trials is more and
(
)
more closely approximated by N m n, s n as n grows larger.
Second Version of CLT: Let X be a random variable with mean m and standard
deviation s . The distribution for the average of the outputs of X over n trials is more and
more closely approximated by N m, sn as n grows larger.
(
)
Examples
A Neilson report stated the children between the ages of 2 and 5 watch an average of 25
hours of television per week. Assume this variable is normally distributed with a standard
deviation of 3 hours.
1. If 20 children between the ages of 2 and 5 are randomly selected, what is the
probability that total number of hours of television they watch per week is between
490 and 520 hours?
2. If 20 children between the ages of 2 and 5 are randomly selected, what is the
probability that they watch an average of 26.3 hours of television per week?
Reminder: A z-score, or standardized score, converts a value from a normal
distribution with mean m and standard deviation s into a value on the unit normal
distribution (with mean 0 and standard deviation 1). In other words, it indicates how
many standard deviations above (or below) the mean a particular value falls.
x-m
z - score =
s
Homework
1. A special number cube has a mean of 5 and standard deviation of 2 for one roll.
a. The number cube is rolled 20 times and the results added together. What is the
probability that the sum is exactly 90?
b. The number cube is rolled 20 times and the results added together. What is the
probability that the sum is between 80 and 110?
c. The number cube is rolled 20 times and the results averaged. What is the
probability that the average is between 4.5 and 7?
2. Suppose that a college admissions office needs to compare scores of students who
take the SAT (Scholastic Aptitude Test) with those who take the ACT (American
College Test). Suppose that among the college’s applicants who take the SAT, scores
have a mean of 1440 and a standard deviation of 261. Further suppose that among
the college’s applicants who take the ACT, scores have a mean of 20.6 and a standard
deviation of 5.2. If applicant Bobby scored 1620 on the SAT and applicant Kathy
scored 28 on the ACT, who performed better? Use z-scores to support your answer.
3. You roll a number cube 360 times and count the number of twos. Is it unusual to only
get 35 twos? Use the z-score to decide.
4. Sara and Becca are very competitive. They both run track but compete in different
races. Sara ran the 100-yard dash and had a time of 10.5 seconds where the overall
mean was 11 sec and standard deviation was 0.4 seconds. Becca ran the 1-mile and
had a time of 6.8 minutes where the overall mean was 7.3 minutes and standard
deviation was .6 minutes. Who performed better? Use z-scores to support your
answer.
5. For a certain brand of light bulbs, each bulb has a mean life expectancy of 675 hours
and a standard deviation of 50 hours.
a. What is the probability of a case of 12 light bulbs lasting an average of 660 hours?
b. What is the probability of a case of 12 light bulbs lasting an average of between
670 and 700 hours?
c. What is the probability of a case of 12 light bulbs lasting a total of between 8,000
and 8,300 hours?
6. A 50-question multiple-choice test has 5 options for each question.
a. If someone randomly guesses on each question, what is the probability that they
get a score between 8 and 11?
b. Twenty people took this test and randomly guessed on each question. What is the
probability that their average score is between 8 and 11?
Answers
1. a. 0.024 b. 0.856 c. 0.868
2. Kathy
3. unusual
4. Sara
5. a. 0.016 b. 0.594 c. 0.594
6. a. 0.398 b. 0.942